Notice that, by integration by parts, , where is solution of the adjoint equation:

Sparse control: ()

The stationary problem

Optimality conditions

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a prox-prox splitting: first write the state as , then

Proximal-point step:

Proximal-point step:

Remark: Notice that, when , the solution of is simply given by

Evolutionary problem

Optimality conditions

Define the classical Lagrangian

By integration by parts, we have

Deriving with respect to the three variables , we obtain the optimality system:

where the relation between the optimal control and the dual state is given by

The latter is equivalent to

where the operator of is defined by

Finally,

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a grad-prox splitting:

Gradient step:

where

and

Proximal-point step:

Remarks: Another possibility is to include the term in the proximal step.
Notice that, for

then is Lipschitz continuous. Indeed, for (), then

where

and

By linearity and solve the same equations with right-hand-sides and , respectively. Then

where we defined

In order the prox-grad method to converge, the restriction on the step size is given by

Sparse state: ()

The stationary problem

Optimality conditions

Finally, we obtain a single equation in the dual variable :

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a prox-prox splitting on the Augmented Energy: first write the state as , then

Proximal-point step:

Proximal-point step:

where we defined

Remark: Notice that again, when , the solution of is simply given by

Evolutionary problem

Optimality conditions

Define the classical Lagrangian

By integration by parts, we have

Deriving with respect to the three variables , we obtain the optimality system:

where the relation between the optimal control and the dual state is given by

The adjoint equation is equivalent to

Finally,

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a grad-prox splitting on the following Augmented Energy:

Then,

Gradient step:

where

and

Proximal-point step:

where we defined

Remark: Another possibility is to consider

Computational experiments

In the following, we present the setting for the numerical experiments.

Spacial domain: ;

Time interval: , with ;

Weight-parameters: , , and ;

Trajectory target:

where , ;

Control operator: for and ,

is the finite difference discretization of ;

Numerical grid: in space, in time.

Stationary solutions

Figure 1: .

Figure 2: , .

Figure 3: , .

Evolutionary problem

Optimal control

Figure 4a: Optimal control for . In red, the controllable subdomain; in blue, the stationary optimal controls.Figure 4b: Optimal control for , . In red, the controllable subdomain; in blue, the stationary optimal controls.Figure 4c: Optimal control for , . In red, the controllable subdomain; in blue, the stationary optimal controls.

Optimal state

Figure 5a: Optimal state for . In red, the target ; in blue, the stationary optimal states.Figure 5b: Optimal state for , . In red, the target ; in blue, the stationary optimal states.Figure 5c: Optimal state for , . In red, the target ; in blue, the stationary optimal states.